Tuesday, April 18, 2017

All of string theory's power, beauty depends on quantum mechanics

Wednesday papers:Arkani-Hamed et al. show that the amplituhedron is all about sign flips. Maldacena et al. study the double-trace deformations that make a wormhole traversable. Among other things, they argue that the cloning is avoided because the extraction (by "Bob") eliminates the interior copy of the quantum information.

String/M-theory is the most beautiful, powerful, and predictive theory we know – and, most likely, the #1 with these adjectives among those that are mathematically possible – but the degree of one's appreciation for its exceptional credentials depends on one's general knowledge of physics, especially quantum mechanics.

Quantum mechanics was basically discovered at one point in the mid 1920s and forced physics to make a one-time quantum jump. On the other hand, it also defines a trend because the novelties of quantum mechanics may be taken more or less seriously, exploited more or less cleverly and completely, and as physics was evolving towards more advanced, stringy theories and explanations of things, the role of the quantum mechanical thinking was undoubtedly increasing.

When we say "classical string theory", it is a slightly ambiguous term. We can take various classical limits of various theories that emerge from string theory, e.g. the classical field theory limit of some effective field theories in the spacetime. But the most typical representation of "classical string theory" is given by the dull yellow animation above. A classical string is literally a curve in a pre-existing spacetime that oscillates according to a wave equation of a sort.

OK, on that picture, you see a vibrating rope. It is not better or more exceptional than an oscillating membrane, a Chladni pattern, a little green man with Parkinson's disease, or anything else that moves and jiggles. The power of string theory only emerges once you consider the real, adult theory where all the observables such as the positions of points along the string are given by non-commuting operators.

Just to be sure, the rule that "observable = measurable quantities are associated with non-commuting operators" is what I mean by quantum mechanics.

What does quantum mechanics do for a humble string like the yellow string above?

First, it makes the spectrum of vibrations discrete.

Classically, you may change the initial state of the vibrating string arbitrarily and continuously, and the energy carried by the string is therefore continuous, too. That's not the case in quantum mechanics. Quantum mechanics got its name from the quantized, discrete eigenvalues of the energy. A vibrating string is basically equivalent to a collection of infinitely many harmonic oscillators. Each quantum mechanical harmonic oscillator only carries an integer number of excitations, not a continuous amount of energy.

The discreteness of the spectrum – which depends on quantum mechanics for understandable reasons – is obviously needed for strings in string theory to coincide with a finite number of particle species we know in particle physics – or a countable one that we may know in the future. Without the quantization, the number of species would be uncountably infinite. The species would form a continuum. There would be not just an electron and a muon but also elemuon and all other things in between, in an infinite-dimensional space.

Quantum mechanics is needed for some vibrating strings to act as gravitons and other exceptional particles.

String theory predicts gravity. It makes Einstein's general relativity – and the curved spacetime and gravitational waves that result from it – unavoidable. Why is it so? It's because some of the low-energy vibrating strings, when they're added into the spacetime, have exactly the same effect as a deformation of the underlying geometry – or other low-energy fields defining the background.

Why is it so? It's ultimately because of the state-operator correspondence. The internal dynamics of a string depends on the underlying spacetime geometry. And the spacetime geometry may be changed. But the infinitesimal change of the action etc. for a string is equivalent to the interaction of the string with another, "tiny" string that is equivalent to the geometry change.

We may determine the right vibration of the "tiny" string that makes the previous sentence work because for every operator on the world sheet (2D history of a fundamental string), there exists a state of the string in the Hilbert space of the stringy vibrations. And this state-operator correspondence totally depends on quantum mechanics, too.

In classical physics, the number of observables – any function \(f(x_i,p_i)\) on a phase space – is vastly greater than the number of states. The states are just points given by the coordinates \((x_i,p_i)\) themselves. It's not hard to see that the first set is much greater – an infinite-dimensional vector space – than the second. However, quantum mechanics increases the number of states (by allowing all the superpositions) and reduces the number of observables (by making them quantized, or respectful towards the quantization of the phase space) and the two numbers become equivalent up to a simple tensoring with the functions of the parameter \(\sigma\) along the string.

I don't want to explain the state-operator correspondence, other blog posts have tried it and it is a rather technical issue in conformal field theory that you should study once you are really serious about learning string theory. But here, I want to emphasize that it wouldn't be possible in any classical world.

Let me point out that the world of the "interpreters" of quantum mechanics who imagine that the wave function is on par with a classical wave is a classical world, so it is exactly as impotent as any other world.

T-duality depends on quantum mechanics

A nice elementary symmetry that you discover in string theory compactified on tori is the so-called T-duality. The compactified string theory on a circle of radius \(R\) is the same as the theory on a circle of radius \(\alpha' / R\) where \(T=1/2 \pi \alpha'\) is the string tension (energy or mass per unit length of the string). Well, this property depends on quantum mechanics as well because the T-duality map exchanges the momentum \(n\) with the winding \(w\) which are two integers.

But in a classical string theory, the winding number \(w\in \ZZ\) would still be integer (it counts how many times a closed string is wrapped around the circle) while the momentum would be continuous, \(n\in\RR\). So they couldn't be related by a permutation symmetry. The T-duality couldn't exist.

The fancier features of string theory you look at, the more obviously unavoidable quantum mechanics becomes. One of the funny things of bosonic string theory compactified on a circle is that the generic gauge group \(U(1)\times U(1)\) gets enhanced to \(SU(2)\times SU(2)\) on the self-dual radius. Even though you start with a theory where everything is "Abelian" or "linear" in some simple sense – a string propagating on a circle – you discover that the non-Abelian \(SU(2)\) automatically arises if the radius obeys \(R = \alpha' / R\), if it is self-dual.

First, one must understand that for a generic radius, the unbroken gauge group is \(U(1)\times U(1)\). One gets two \(U(1)\) gauge groups because the gauge fields are basically \(g_{\mu,25}\) and \(B_{\mu,25}\). They arise as "last columns" of a symmetric tensor, the metric tensor, and an antisymmetric tensor, the \(B\)-field. The first (metric tensor-based) \(U(1)\) group is the standard Kaluza-Klein gauge group and it is \(U(1)\) because \(U(1)\) is the isometry group of the compactification manifold. There is another gauge group arising from the gauge field that you get from a pre-existing 2-index gauge field \(B_{\mu\nu}\), a two-form, if you set the second index equal to the compactified direction.

These two gauge fields are permuted by the T-duality symmetry (just like the momentum and winding are permuted, because the momentum and winding are really the charges under these two symmetries).

OK, how do you get the \(SU(2)\)? The funny thing is that the \(U(1)\) gauge bosons are associated, via the operator-state correspondence mentioned above, with the operators on the world sheet\[

(\partial_z X^{25}, \quad \partial_{\bar z} X^{25}).

\] One of them is holomorphic, the other one is anti-holomorphic, we say. T-duality maps these operators to\[

(\partial_z X^{25}, \quad -\partial_{\bar z} X^{25}).

\] so it may be understood as a mirror reflection of the \(X^{25}\) coordinate of the spacetime except that it only acts on the anti-holomorphic (or right-moving) oscillations propagating along the string. That's great. You have something like a discrete T-duality which is just some sign flip or, equivalently, the exchange of the momentum and winding. How do you get a continuous \(SU(2)\), I ask again?

The funny thing is that at the self-dual radius, there are not just two operators like that but six. The holomorphic one, \(\partial_z X^{25}\), becomes just one component of a three-dimensional vector\[

(\partial_z X_L^{25},\,\, :\exp(+i X_L^{25}):, :\exp(-i X_L^{25}):)

\] Classically, the first operator looks nothing like the last two. If you have a holomorphic function \(X_L^{25}(z)\) of some coordinate \(z\), its \(z\)-derivative seems to be something completely different than its exponential, right? But quantum mechanically, they are almost the same thing! Why is it so?

If you want to describe all physically meaningful properties of three operators like that, the algebra of all their commutators encodes all the information. Just like string theory has the state-operator correspondence that allows you to translate between states and operators, it also has the OPEs – operator-product expansions – that allow you to extract the commutators of operators from the singularities in a decomposition of their products etc.

And it just happens that the singularities in the OPEs of any such operators are compatible with the statement that these three operators are components of a triplet that transforms under an \(SU(2)\) symmetry. So you get one \(SU(2)\) from the left-moving, \(z\)-dependent part \(X_L^{25}\), and one \(SU(2)\) from the \(\bar z\)-dependent \(X_R^{25}\).

All other non-Abelian and sporadic or otherwise cool groups that you get from perturbative string theory arise similarly, and are therefore similarly dependent on quantum mechanics. For example, the monster group in the string theory model explaining the monstrous moonshine only exists because of a similar "equivalence" that is only true at the quantum level.

Spacetime dimension and sizes of group are only predictable in quantum mechanics

String theory is so predictive that it forces you to choose a preferred dimension of the spacetime. The simple bosonic string theory has \(D=26\) and superstring theory, the more realistic and fancy one, similarly demands \(D=10\). This contrasts with the relatively unconstrained, "anything goes" theories of the pre-stringy era.

Polchinski's book contains "seven" ways to calculate the critical dimension, according to the counting by the author. But here, what is important is that all of them depend on a cancellation of some quantum anomalies.

In the covariant quantization, \(D=26\) basically arises as the number of bosonic fields \(X^\mu\) whose conformal anomaly cancels that from the \(bc\) ghost system. The latter has \(c=1-3k^2=-26\) because some constant is \(k=3\): the central charge describes a coefficient in front of a standard term to the conformal anomaly. Well, you need to add \(c=+26\) – from 26 bosons – to get zero. And you need to get zero for the conformal symmetry to hold, even in the quantum theory. And the conformal symmetry is needed for the state-operator correspondence and other things – it is a basic axioms of covariant perturbative string theory.

Alternatively, you may define string theory in the light-cone gauge. The full Lorentz symmetry won't be obvious anymore. You will find out that some commutators\[

[j^{i-},j^{j-}] = \dots

\] in the light-cone coordinates behaves almost correctly. Except that when you substitute the "bilinear in stringy oscillators" expressions for the generators \(j^{i-}\), the calculation of the commutator will contain not only the "single contractions" – this part of the calculation is basically copying a classical calculation – but also the "double contraction" terms. And those don't trivially cancel. You will find out that they only cancel for 24 transverse coordinates. Needless to say, the "double contraction" is something invisible at the level of the Poisson brackets. You really need to talk about the "full commutators" – and therefore full quantum mechanics, not just some Poisson-bracket-like approximation – to get these terms at all.

Again, the correct spacetime dimension \(D=26\) or \(D=10\) arises from the cancellation of some quantum anomaly – some new quantum mechanical effects that have the potential of spoiling some symmetries that "trivially" hold in the classical limit that may have inspired you. The prediction couldn't be there if you ignored quantum mechanics.

The field equations in the spacetime result from an anomaly cancellation, too.

If you order perturbative strings to propagate on a curved spacetime background, you may derive Einstein's equations (plus stringy short-distance corrections), which in the vacuum simply demand the Ricci-flatness \[

R_{\mu\nu} = 0.

\] A century ago, Einstein had to discover that this is what the geometry has to obey in the vacuum. It's an elegant equation and among similarly simple ones, it's basically unique that is diffeomorphism-symmetric. And you may derive it from the extremization of the Einstein-Hilbert action, too.

However, string theory is capable of doing all this guesswork for you. In other words, string theory is capable of replacing Einstein's 10 years of work. You may derive the Ricci-flatness from the cancellation of the conformal anomaly, too. You need the world sheet theory to stay invariant under the scaling of the world sheet coordinates, even at the quantum level.

But the world sheet theory depends on the functions\[

g_{\mu\nu} (X^\lambda(\sigma,\tau))

\] and for every point in the spacetime given by the numbers \(\{X^\lambda\}\), you have a whole symmetric tensor \(g_{\mu\nu}\) of parameters that behave like "coupling constants" in the theory. But in a quantum field theory, and the world sheet theory is a quantum field theory, every coupling constant generically "runs". Its value depends on the chosen energy scale \(E\). And the derivative with respect to the scale\[

\frac{dg_{\mu\nu}(X^\lambda)}{d (\ln E)} = \beta_{\mu\nu}(X^\lambda)

\] is known as the beta-function. Here you have as many beta-functions as you have the numbers that determine the metric tensor at each spacetime point. The beta-functions have to vanish for the theory to remain scale-invariant on the world sheet – and you need it. And you will find out that\[

\beta_{\mu\nu}(X^\lambda) = R_{\mu\nu} (X^\lambda).

\] The beta-function is nothing else than the Ricci tensor. Well, it could be the Einstein tensor and there could be extra constants and corrections. But I want to please you with the cool stuff; I hope that you don't doubt that if you want to work with these things, you have to take care of many details that make the exact answers deviate from the most elegant, naive Ansatz with the given amount of beauty.

So Einstein's equations result from the cancellation of the conformal anomaly as well. The very requirement that the theory remains consistent at the quantum level – and the preservation of gauge symmetries is indeed needed for the consistency – is enough to derive the equations for the metric tensor in the spacetime.

Needless to say, this rule generalizes to all the fields that you may get from particular vibrating strings in the spacetime. Dirac, Weyl, Maxwell, Yang-Mills, Proca, Higgs, and other equations of motions for the fields in the spacetime (including all their desirable interactions) may be derived from the scale-invariance of the world sheet theory, too.

In this sense, the logical consistency of the quantum mechanical theory dictates not only the right spacetime dimension and other numbers of degrees of freedom, sizes of groups such as \(E_8\times E_8\) or \(SO(32)\) for the heterotic string (the rank must be \(16\) and the dimension has to be \(496\), among other conditions), but the consistency also determines all the dynamical equations of motion.

S-duality, T-duality, mirror symmetry, AdS/CFT and holography, ER-EPR, and so on

And I could continue. S-duality – the symmetry of the theories under the \(g\to 1/g\) maps of the coupling constant – also depend on quantum mechanics. It's absolutely obvious that no S-duality could ever work in a classical world, not even in quantum field theory. Among other things, S-dualities exchange the elementary electrically charged particles such as electrons with the magnetically charged ones, the magnetic monopoles. But classically, those are very different: electrons are point-like objects with an "intrinsic" charge while the magnetic monopoles are solitonic solutions where the charge is spread over the solution and quantized because of topological considerations.

However, quantum mechanically, they may be related by a permutation symmetry.

Mirror symmetry is an application of T-duality in the Calabi-Yau context, so everything I said about the quantum mechanical dependence of T-duality obviously holds for mirror symmetry, too.

Holography in quantum gravity – as seen in AdS/CFT and elsewhere – obviously depends on quantum mechanics, too. The extra holographic dimension morally arises from the "energy scale" in the boundary theory. But the AdS space has an isometry relating all these dimensions. Classically, "energy scale" cannot be indistinguishable from a "spacetime coordinate". Classically, the energy and momentum live in a spacetime, they have different roles.

Quantum mechanically, there may be such symmetries between energy/momentum and position/timing. The harmonic oscillator is a basic template for such a symmetry: \(x\) and \(p\) may be rotated to each other.

ER-EPR talks about the quantum entanglement so it's obvious that it would be impossible in a classical world.

I could make the same point about basically anything that is attractive about string theory – and even about comparably but less intriguing features of quantum field theories. All these things depend on quantum mechanics. They would be impossible in a classical world.

Quantum mechanics does a lot of things. You have seen many examples – and there are many others – that quantum mechanics generally allows you to find symmetries between objects that look classically totally different. Like the momentum and winding of a string. Or the derivative of \(X\) with the exponential of \(X\) – at the self-dual radius. Or the states and operators. Or elementary particles and composite objects such as magnetic monopoles. And so on, and so on.

Sometimes, the spectrum of a quantity becomes discrete in order for the map or symmetry to be possible.

Sometimes, just the qualitative differences are erased. Sometimes, all the differences are erased and quantum mechanics enables the emergence of exact new symmetries that would be totally crazy within classical physics. Sometimes, these symmetries are combined with some naive ones that already exist classically. \(U(1)\times U(1)\) may be extended to \(SU(2)\times SU(2)\) quantum mechanically. Similarly, \(SO(16)\times SO(16)\) in the fermionic definition or \(U(1)^{16}\) in the bosonic formulation of the heterotic string gets extended to \(E_8\times E_8\). A much smaller, classically visible discrete group gets extended to the monster group in the full quantum string theory explaining the monstrous moonshine.

Whenever a classical theory would be getting dangerously singular, quantum mechanics changes the situation so that either the dangerous states disappear or they're supplemented with new degrees of freedom or another cure. In many typical cases, the "potentially dangerous regime" of a theory – where you could be afraid of an inconsistency – is protected and consistent because quantum mechanics makes all the modifications and additions needed for that regime to be exactly equivalent to another theory that you have known – or whose classical limit you have encountered. Quantum mechanics is what allows all the dualities and the continuous connection of all seemingly inequivalent vacua of string/M-theory into one master theory.

All the constraints - on the number of dimensions, sizes of gauge groups, and even equations of motion for the fields in spacetime – arise from the quantum mechanical consistency, e.g. from the anomaly cancellation conditions.

When you become familiar with all these amazing effects of string theory and others, you are forced to start to think quantum mechanically. You will understand that the interesting theory – with the uniqueness, predictive power, consistency, symmetries, unification of concepts – is unavoidably just the quantum mechanical one. There is really no cool classical theory. The classical theories that you encounter anywhere in string theory are the classical limits of the full theory.

You will unavoidably get rid of the bad habit of thinking of a classical theory as the "primary one", while the quantum mechanical theory is often considered "derived" from it by the beginners (including permanent beginners). Within string/M-theory, it's spectacularly clear that the right relationship is going in the opposite direction. The quantum mechanical theory – with its quantum rules, objects, statements, and relationships – is the primary one while classical theories are just approximations and caricatures that lack the full glory of the quantum mechanical theory.